Mathematics: Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. f of x equals four divided by x. and g of x equals four divided by x

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Confirm that f and g are inverses by showing

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alexkrol10

30.05.2020

Mathematics

Given f(x) = ⁴/ₓ and g(x) = ⁴/ₓ.

$\boxed{ \ (f \circ g)(x) = f(g(x)) \ }$

$\boxed{ \ f(g(x)) = \frac{4}{\Big(\frac{4}{x}\Big)} \ }$

$\boxed{ \ f(g(x)) = 4 \times \frac{x}{4}} \ }$

$\boxed{\boxed{ \ f(g(x)) = x \ }}$

$\boxed{ \ (g \circ f)(x) = g(f(x)) \ }$

$\boxed{ \ g(f(x)) = \frac{4}{\Big(\frac{4}{x}\Big)} \ }$

$\boxed{ \ g(f(x)) = 4 \times \frac{x}{4}} \ }$

$\boxed{\boxed{ \ g(f(x)) = x \ }}$

Based on the work steps above, it appears that $\boxed{\boxed{ \ f (g (x)) = x \ and \ g (f (x)) = x, \ }}$ thus it has been confirmed that f and g are inverses.

Further explanation

Part-A

$\boxed{ \ f(x) = \frac{4}{x} \rightarrow f^{-1}=? \ }$

Let $\boxed{ \ y = \frac{4}{x} \ }$

Rewrite and set the function with subject x.

Both sides are divided by x.

$\boxed{ \ xy = 4 \ }$

Both sides are divided by y.

$\boxed{ \ x = \frac{4}{y} \ }$

Let's replace $\boxed{ \ x \ with \ f^{-1}(x) \ }$ and $\boxed{ \ y \ with \ x \ }$ .

Thus, the inverse of $\boxed{ \ f(x) = \frac{4}{x} \ is \ \boxed{ \ f^{-1} = \frac{4}{x} \ }}$

Part-B

$\boxed{ \ g(x) = \frac{4}{x} \rightarrow g^{-1}=? \ }$

Let $\boxed{ \ y = \frac{4}{x} \ }$

Rewrite and set the function with subject x.

Both sides are divided by x.

$\boxed{ \ xy = 4 \ }$

Both sides are divided by y.

$\boxed{ \ x = \frac{4}{y} \ }$

Let's replace $\boxed{ \ x \ with \ f^{-1}(x) \ }$ and $\boxed{ \ y \ with \ x \ }$ .

Thus, the inverse of $\boxed{ \ g(x) = \frac{4}{x} \ is \ \boxed{ \ g^{-1} = \frac{4}{x} \ }}$

Notes:

A function has an inverse if, and only if, it is one-to-one and onto.Let f be one-to-one and onto function with the domain A and the range B. Then its inverse function $\boxed{ \ f^{-1} \ }$ has the domain B and the range A such that $\boxed{ \ f(x) = y \leftrightarrow f^{-1}(y) = x \ }.$ By definition the inverse function $f^{-1} \ undoes \ what \ f \ does.$ That is, if we take

apply

and then apply $f^{-1},$ we arrive back at

where we started. Similarly, $f \ undoes \ what \ f^{-1} \ does.$ That is why $f \ and \ f^{-1}$ are the inverses of each other.

$\boxed{\boxed{ \ f(f^{-1}(x)) = x \ \ and \ \ f^{-1}(f(x)) = x \ }}$

To find the inverse, we use the same procedures that we used for relations. Drawing the reflection with respect to $\boxed{ \ y=x \ }$ we get the following picture as in the attachment.Learn more Determine the following pairs of functions that inverse of each other. The inverse of a function If g(x) is the inverse of f(x), what is the value of f(g(2))

Keywords: the inverse of function, confirm that f and g are inverses, by showing that f(g(x)) = x, g(f(x)) = x, four divided by x, one-to-one, onto

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. f of x equals four di

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Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. f of x equals four divided by x. and g of x equals four divided by x

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